AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
over 
. Ignore "pathological" solutions.[/quote]
are positive real numbers such that 
, Prove that ![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
of the triangle are taken points 
respctively so that 
. 
, 
. The area of triangle is equal to 
. Calculate the area of triangle 
.
![$[CGB]=z.S$](http://latex.artofproblemsolving.com/8/f/7/8f724137f0507f0fad44dceff9644f3e2f0a8a84.png)
; ![$[GFC]=(1-y)z.S$](http://latex.artofproblemsolving.com/1/e/8/1e8d470eb67ced9cd901f5b9b1ef7f8c98cb9430.png)
. Similarly, ![$[CEA]=x.S$](http://latex.artofproblemsolving.com/c/4/3/c43a9b77c53357b4a6f6fe1229be14b4b9c5aa62.png)
; ![$[AEG]=(1-z)x.S$](http://latex.artofproblemsolving.com/d/9/8/d98f1aa15c1a9f2b808a6b851218be722bc138ea.png)
. By symmetry, ![$[EBF]=(1-x)y.S$](http://latex.artofproblemsolving.com/c/8/c/c8c7fd8f921c7c0d00b710a34dd458cd4105aa4d.png)
. Thus ![$$[EGF]=S[1-(1-y)z-(1-z)x-(1-x)y] = S[1-(x+y+z)+(xy+yz+xx)]$$](http://latex.artofproblemsolving.com/b/9/e/b9e68fc16bf8f56be2ae55127b1e3508f76c23f5.png)
.
and 
.
